Question

simplify= 2√6/√2+√3 +6√2/√6+√3 -8√3/√6+√2.

Answer 1

Hey friend,
Here is the answer you were looking for:
$\frac{2 \sqrt{6} }{ \sqrt{2} + \sqrt{3} } + \frac{6 \sqrt{2} }{ \sqrt{6} + \sqrt{3} } - \frac{8 \sqrt{3} }{ \sqrt{6} + \sqrt{2} } \\ \\ on \: rationalizing \: we \: get \\ \\ = \frac{2 \sqrt{6} }{ \sqrt{2} + \sqrt{3} } \times \frac{ \sqrt{2} - \sqrt{3} }{ \sqrt{2} - \sqrt{3} } + \frac{6 \sqrt{2} } { \sqrt{6} + \sqrt{3} } \\ \times \frac{ \sqrt{6} - \sqrt{3} }{ \sqrt{6} - \sqrt{3} } - \frac{8 \sqrt{3} }{ \sqrt{6} + \sqrt{2} } \times \frac{ \sqrt{6} - \sqrt{2} }{ \sqrt{6} - \sqrt{2} } \\ \\ using \: the \: identity \\ (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ = \frac{2 \sqrt{6} \times \sqrt{2} - 2 \sqrt{6} \times \sqrt{3} }{ {( \sqrt{2}) }^{2} - {( \sqrt{3} )}^{2} } + \frac{6 \sqrt{2} \times \sqrt{6} - 6 \sqrt{2} \times \sqrt{3} }{ {( \sqrt{6}) }^{2} - {( \sqrt{3}) }^{2} } - \frac{8 \sqrt{3} \times \sqrt{6} - 8 \sqrt{3} \times \sqrt{2} }{ {( \sqrt{6}) }^{2} - {( \sqrt{2} )}^{2} } \\ \\ = \frac{2 \sqrt{12} - 2 \sqrt{18} }{2 - 3} + \frac{6 \sqrt{12} - 6 \sqrt{6} }{6 - 3} - \frac{8 \sqrt{18} - 8 \sqrt{6} }{6 - 2} \\ \\ = \frac{2 \times 2 \sqrt{3} - 2 \times 3 \sqrt{2} }{ - 1} + \frac{6 \times 2 \sqrt{3} - 6 \sqrt{6} }{3} - \frac{8 \times 3 \sqrt{2} - 8 \sqrt{6} }{4} \\ \\ = - 4 \sqrt{3} + 6 \sqrt{2} + \frac{12 \sqrt{3} - 6 \sqrt{6} }{3} - \frac{24 \sqrt{2} - 8 \sqrt{6} }{4} \\ \\ = \frac{ - 4 \sqrt{3} \times 12 + 6 \sqrt{2} \times 12 + 12 \sqrt{3} \times 4 - 6 \sqrt{6} \times 4 - (24 \sqrt{2} \times 3 - 8 \sqrt{6} \times 3}{12} \\ \\ = \frac{ - 48 \sqrt{3} + 72 \sqrt{2} + 48 \sqrt{3} - 24 \sqrt{6} - 72 \sqrt{2} + 32 \sqrt{6} }{12} \\ \\ = \frac{ - 24 \sqrt{6} + 32 \sqrt{6} }{12} \\ \\ = \frac{8 \sqrt{6} }{12} \\ \\ = \frac{2 \sqrt{6} }{3}$

Hope this helps!!!!

@Mahak24

Thanks...
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